By studying, Will can produce a higher grade, Gw, on an upcoming economics exam.

Question

By studying, Will can produce a higher grade, Gw, on an upcoming economics exam. His production function depends on number of hour he studies marginal analysis problems, A, and the number of hours he studies supply and demand problems, R. Specifically, Gw=2.5A^.36(R^.64). His roommate David's grade production function is Gd=2.5A^.25(R^.75).
a. What is Will's marginal productivity from studying supply and demand problems (R) problems? David's?
b. What is Will's marginal rate of technical Substituion between studying the two types of problems? David's?
c. Is it possible that Will and David have different marginal productivity functions but the same marginal rate of technical substitution functions? Explain By studying, Will can produce a higher grade, Gw, on an upcoming economics exam. His production function depends on number of hour he studies marginal analysis problems, A, and the number of hours he studies supply and demand problems, R. Specifically, Gw=2.5A^.36(R^.64). His roommate David's grade production function is Gd=2.5A^.25(R^.75).
a. What is Will's marginal productivity from studying supply and demand problems (R) problems? David's?
b. What is Will's marginal rate of technical Substituion between studying the two types of problems? David's?
c. Is it possible that Will and David have different marginal productivity functions but the same marginal rate of technical substitution functions? Explain
a. What is Will's marginal productivity from studying supply and demand problems (R) problems? David's?
b. What is Will's marginal rate of technical Substituion between studying the two types of problems? David's?
c. Is it possible that Will and David have different marginal productivity functions but the same marginal rate of technical substitution functions? Explain

Explanation / Answer

a.
Gw = 2.5(A)^(0.36)B^(0.64)
Take the derivative with respect to B.
Will's MPB =2.5*0.64*(A/B)^0.36
Will's MPB =1.6*(A/B)^0.36

Gd=2.5A^(0.25)B^(0.75)
Take the derivative with respect to B.
David's MPB = 2.5*0.75*(A/B)^0.25
David's MPB = 1.875*(A/B)^0.25

b.
MRTS = MPA/MPB

Gw = 2.5(A)^(0.36)B^(0.64)
Take the derivative with respect to A.
Will's MPA = 2.5*0.36*(B/A)^(0.64)
Will's MPA = 0.9*(B/A)^(0.64)
Will's MRTS = MPA/MPB
Will's MRTS = 0.9*(B/A)^(0.64)/1.6*(A/B)^0.36
Will's MRTS = 0.9*(B/A)/1.6
Will's MRTS = (9/16)*(B/A)

Gd=2.5A^(0.25)B^(0.75)
Take the derivative with respect to A.
David's MPA = 2.5*0.25*(B/A)^(0.75)
David's MPA = 0.625*(B/A)^(0.75)
David's MRTS = MPA/MPB
David's MRTS = 0.625*(B/A)^(0.75)/1.875*(A/B)^0.25
David's MRTS = 0.625*(B/A)/1.875
David's MRTS = (1/3)*(B/A)

c.
Yes. All you need are different scalar multiples. For example, let's say Will had Gw = C*(A)^(p)B^(q) and David had Gd = D*(A)^(p)B^(q) where C does not equal D. They would have different marginal productivity function but the same MRTS

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